Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Sel -Made Algeb aic Magic Squa es o O de 11
Inde J. Taneja1
The whole wo k as pd iles is a ailable a au ho ’s si es:
h ps://numbe s-magic.com/?p=16759
This wo k is wi hou use o any kind o p og amming language
Abs ac
This wo k b ings magic squa es o o de 11 o educed en ies. By educed o less en ies, we unde s and ha ins ead o
no mal n2en ies o a magic squa e o de n, we a e using less numbe o en ies. Mo eo e , in hese si ua ions he en ies a e
no mo e sequen ial numbe s. These en ies a e non-sequen ial posi i e and nega i e numbe s. Some imes, we call hese
kind o magic squa es as sel -made. I means ha hese a e comple e in hemsel es. Jus pu he alues o en ies and choose
he magic sum, we ge a magic squa e. In some cases, he e maybe decimal o ac ional alues o en ies depending on he
ypes o magic squa es. Diffe en kind o magic squa es a e used o b ing hese sel -made magic squa es. These a e o ype,
block-wise,co ne ed,single-digi bo de ed,double-digi bo de ed, e c. In some cases, he idea o magic ec angles is
also applied. In each case, he magic ec angles a e conside ed wi h equal wid h and leng h. This wo k o he sel -made
algeb aic magic squa es o de 11. This wo k is a ailable online a abo e gi en link. Fo simila kind o wo k o diffe en o de s
he eade s a e sugges ed o au ho ’s wo k gi en in e e ences [30]-[45].
1Fo me ly, P o esso o Ma hema ics, Fede al Uni e si y o San a Ca a ina, Flo ianópolis, SC, B azil (1978-2012).
E-mail: [email p o ec ed]; Web-si es: h p://inde j aneja.wo dp ess.com; h p://numbe s-magic.com;
Twi e : @IJTANEJA; Ins ag am: @c azynumbe s.
1
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Con en s
1 In oduc ion 2
2 Magic Squa es o O de 11 3
2.1 Sel -MadeMagicSqua eso O de 11 ..................................................... 4
3 Au ho ’s Con ibu ion o Magic Squa es and Rec ea ion Numbe s 54
4 Acknowledgemen 55
1 In oduc ion
This wo k b ings sel -made algeb aic magic squa es o o de 11. By sel -made o educed o less en ies, we unde s and ha
ins ead o no mal n2en ies o a magic squa e o de n, we a e using less numbe s o en ies, whe e he magic squa e is comple e in
i sel . Pu ing any in ege alues o hese less en ies, we shall ge always a magic squa e. Mo eo e , in hese si ua ions he en ies
a e no mo e sequen ial numbe s. These en ies a e non-sequen ial posi i e and/o nega i e numbe s. In some cases, hese may
be decimal o ac ional alues depending on he way o choosing he en ies. Some ime o a oid decimal o ac ional en ies we
apply ce ain condi ions. These condi ions depends on he ypes o magic squa es. The name sel -made is no known in he li e a u e o
magic squa es. I is being in oduced o he i s ime. The wo k is based on diffe en ypes o magic squa es, such as, pandiagonal,
block-wise, co ne ed, single-digi bo de ed, double-digi bo de ed, e c. I is no necessa y, bu we wo ked wi h magic ec angles
ha ing equal wid h and leng h o he same ca ego y wi hin a magic squa e. I we elax his condi ion, i.e., by conside ing only equali y
o wid h, s ill we ha e good esul s.
This wo k o he o de 11 b ings magic,semi-magic and pandiagonal magic squa es. I is di ided in h ee pa s. The i s pa
only on magic squa es, he second pa on semi-magic squa es and he hi d pa on pandiagonal magic squa es o o de 11. This is
he only i s pa . Fo he second pa e e he au ho ’s wo k [43]. Fo mo e de ails on hese kind o wo k e e au ho ’s p e ious wo ks
[30]-[45]. The able below gi e he quan i ies and e e ences o each o de .
2
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
O de Magic Squa es Semi-Magic Squa es Pandiagonal Magic Squa es To al
31 1 0 2[38]
42 0 2 4[38]
52 1 3 6[38]
65 1 6 12 [38]
75 3 8 16 [38]
88 5 13 26 [39]
910 9 18 28 [40]
10 15 14 29 58 [41]
11 25 23 o be done 48 [42, 43]
12 28 25 o be done 53 [44, 45]
The au ho [30, 31, 32, 33, 34, 35] also wo ked simila kind o wo k bu om diffe en poin o iew. This wo k is o he magic squa es
o o de s 3 o 12 o he da es and days o he yea 2025, whe e he da es a e ew en ies and days a e he sums o he magic squa es.
2 Magic Squa es o O de 11
Below a e h ee diffe en examples o magic squa es o o de 11 o sequen ial en ies om 1 o 121.
3
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
The i s example is known by co ne ed magic squa es. The second example is amous as single-digi bo de ed magic squa e. The
hi d example is known as double-digi bo e ed magic squa e. Fo mo e de ails on hese kind o magic squa es e e au ho ’s wo k
[24, 25, 26, 27].
2.1 Sel -Made Magic Squa es o O de 11
Below a e wen y- i e esul s o sel -made magic squa es o o de 11 o educed en ies.
4
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.1. Le ’s conside ollowing sel -made magic squa e o o de 11 wi h educed en ies:
•De ails:
I is a double-digi bo de ed magic squa e o o de 11 embedded wi h a magic squa e o o de 7. The ou magic ec angles
o o de s 2×7a e o equal wid h and leng h. The le e s T and R ep esen s he magic squa es o o de s 7 and 11 espec i ely.
See below an example.
5
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.1. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.1 includes he ollowing magic and semi-magic squa es:
6
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.2. Le ’s conside ollowing sel -made magic squa e o o de 11 wi h educed en ies:
•De ails:
I is a double-digi bo de ed magic squa e o o de 11 embedded wi h ano he double-digi bo de ed magic squa e o o de
7 ha ing magic squa es o o de 3 in he middle. The magic ec angles o o de s 2×7and 2×3a e o equal wid h and leng h in
each case. The le e s M, T and R ep esen s he magic sums o o de s 3, 7 and 11 espec i ely. See below an example.
7
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.2. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.2 includes he ollowing magic and semi-magic squa es:
8
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.3. Le ’s conside ollowing sel -made magic squa e o o de 11 wi h educed en ies:
•De ails:
I is a double-digi bo de ed magic squa e o o de 11 embedded wi h a co ne ed magic squa e o o de 7 ha ing pandiago-
nal magic squa e o o de 5 in he uppe -le co ne . The magic ec angles o o de 2×7and 2×5a e o equal wid h and leng h
in each case. The le e s S, T and R ep esen s he magic sums o o de s 5, 7 and 11 espec i ely. See below an example.
9
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.6. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.6 includes he ollowing magic and semi-magic squa es:
16
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.7. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de 11 ha ing adouble-digi bo de ed magic squa e o o de 7 a he uppe -le co ne . I
again con ains a co ne ed magic squa e o o de 5 wi h magic squa e o o de 3 a he uppe -le co ne . The magic ec angles
o o de s 2×9and 2×5a e o equal wid h and leng h in each case. The le e s M, S, L and R ep esen s he magic sums o
o de s 3, 5, 9 and 11 espec i ely. See below an example.
17
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.7. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.7 includes he ollowing magic and semi-magic squa es:
18
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.8. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de 11 ha ing adouble-digi bo de ed magic squa e o o de 9 a he uppe -le co ne .
I again con ains a single-digi bo de ed magic squa e o o de 5 wi h magic squa e o o de 3 in he inne pa . The magic
ec angles o o de s 2×9and 2×5a e o equal wid h and leng h in each case. The le e s M, S, L and R ep esen s he magic
sums o o de s 3, 5, 9 and 11 espec i ely. See below an example.
19
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.8. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.8 includes he ollowing magic and semi-magic squa es:
20
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.9. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 and 9 ha ing a double-digi bo de ed magic squa e o o de 7 a he uppe -le
co ne . I con ains he magic squa es o o de 3 in he middle. The magic ec angles o o de s 2×3,2×7and 2×9a e o equal
wid h and leng h in each case. The le e s M, T, L and R ep esen s he magic sums o o de s 3, 7, 9 and 11 espec i ely. See
below an example.
21
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.9. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.9 includes he ollowing magic and semi-magic squa es:
22
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.10. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11, 9 and 7 ha ing a pandiagonal magic squa e o o de 5 a he uppe -le co ne . The
magic ec angles o o de s 2×5,2×7and 2×9a e o equal wid h and leng h in each case. The le e s S, T, L and R ep esen s
he magic sums o o de s 5, 7, 9 and 11 espec i ely. See below an example.
23
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.10. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.10 includes he ollowing magic and semi-magic squa es:
24
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.11. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11, 9, 7 and 5 ha ing a magic squa e o o de 3 a he uppe -le co ne . The magic
ec angles o o de s 2×3,2×5,2×7and 2×9a e o equal wid h and leng h in each case. The le e s M, S, T, L and R ep esen s
he magic sums o o de s 3, 5, 7, 9 and 11 espec i ely. See below an example.
25
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.14. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.14 includes he ollowing magic and semi-magic squa es:
32
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.15. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 and 9 ha ing a single-digi bo de ed magic squa e o o de 7 a he uppe -le
co ne . Again i con ains a co ne ed magic squa e o o de 5 ha ing magic squa e o o de 3 a he uppe -le co ne . The
magic ec angles o o de s 2×3,2×7and 2×9a e o equal wid h and leng h in each case. The le e s M, S, T, L and R ep esen s
he magic sums o o de s 3, 5, 7, 9 and 11 espec i ely. See below an example.
33
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.15. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.4 includes he ollowing magic and semi-magic squa es:
34
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.16. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 and 9 ha ing a single-digi bo de ed magic squa e o o de s 7 and 5 a uppe -le
co ne ha ing magic squa e o o de 3 in he middle. The magic ec angles o o de s 2×7and 2×9a e o equal wid h and
leng h in each case. The le e s M, S, T, L and R ep esen s he magic sums o o de s 3, 5, 7, 9 and 11 espec i ely. See below
an example.
35
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.16. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.16 includes he ollowing magic and semi-magic squa es:
36
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.17. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 ha ing a block-wise magic squa e o o de 9 a he uppe -le co ne . I is composed
o 9 equal sums semi-magic squa es o o de 3. The magic ec angles o o de 2×9a e o equal wid h and leng h. The le e s
M L and R ep esen s he magic sums o o de s 3, 9 and 11 espec i ely. In his case T= 3 ×M. See below an example.
37
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.17. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.17 includes he ollowing magic and semi-magic squa es:
38
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.18. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 ha ing a single-digi bo de ed magic squa e o o de 9 a he uppe -le co ne . I
con ains a single-digi bo de ed magic squa e o o de 7 in wi h magic squa e o o de 3 in he middle. The magic ec angles
o o de s 2×9a e o equal wid h and leng h. The le e s M, T, L and R ep esen s he magic sums o o de s 3, 7, 9 and 11
espec i ely. See below an example.
39
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.18. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.18 includes he ollowing magic and semi-magic squa es:
40
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.19. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 ha ing a single-digi bo de ed magic squa e o o de 9 a he uppe -le co ne
wi h magic squa e o o de 7 in he middle. The magic ec angles o o de s 2×9a e o equal wid h and leng h. The le e s T, L
and R ep esen s he magic sums o o de s 7, 9 and 11 espec i ely. See below an example.
41
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.22. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.22 includes he ollowing magic and semi-magic squa es:
48
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.23. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 ha ing a single-digi bo de ed magic squa e o o de 9 a he uppe -le co ne . I
con ains a co ne ed magic squa es o o de s 7 wi h a single-digi bo de ed magic squa e o o de 5 a he uppe -le co ne
wi h magic squa e o o de 3 in he middle. The magic ec angles o o de s 2×5and 2×9a e o equal wid h and leng h in each
case. The le e s M, S, T, L and R ep esen s he magic sums o o de s 3, 5, 7, 9 and 11 espec i ely. See below an example.
49
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.23. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.23 includes he ollowing magic and semi-magic squa es:
50
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.24. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 ha ing a single-digi bo de ed magic squa es o o de s 9 and 7 wi h pandiagonal
magic squa e o o de 5 in he middle. The magic ec angles o o de 2×9a e o equal wid h and leng h. The le e s S, T, L and
R ep esen s he magic sums o o de s 5, 7, 9 and 11 espec i ely. See below an example.
51
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.24. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.24 includes he ollowing magic and semi-magic squa es:
52
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Resul 2.25. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a co ne ed magic squa e o o de s 11 ha ing a single-digi bo de ed magic squa es o o de s 9, 7 and 5 wi h a magic
squa e o o de 3 in he middle. The magic ec angles o o de 2×9a e o equal wid h and leng h. The le e s M, S, T, L and R
ep esen s he magic sums o o de s 3, 5, 7, 9 and 11 espec i ely. See below an example.
53
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
Example 2.25. Le ’s conside an example based on abo e esul :
The magic squa e gi en in Example 2.25 includes he ollowing magic and semi-magic squa es:
3 Au ho ’s Con ibu ion o Magic Squa es and Rec ea ion Numbe s
Fo au ho ’s con ibu ion o magic squa es and ec ea ion numbe s please see he links below:
•Inde J. Taneja, Magic Squa es,
(i) h ps://numbe s-magic.com/?p=668
54
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
(ii) h ps://inde j aneja.wo dp ess.com/2019/06/27/publica ions-magic-squa es/
•Inde J. Taneja, Rec ea ion o Numbe s,
(i) h ps://numbe s-magic.com/?p=671
(ii) h ps://inde j aneja.wo dp ess.com/2019/06/27/publica ions- ec ea ion-o -numbe s/
4 Acknowledgemen
The au ho is hank ul o Mi su oshi Nakamu a o helping in cons uc ion o hese magic squa es, specially in one o he magic squa es
o o de 6.
Re e ences
[1] A. de Winkel, The magic Encyclopedia, h p://home.wanadoo.nl/aaledewinkel/Encyclopedia/index.h ml
[2] C. Boye , Mul imagic Squa es and Cubes, h p://www.mul imagie.com
[3] F. Gaspalou, “Magic Squa es” h p://www.gaspalou. /magic-squa es/
[4] W. T ump,h p://www. ump.de/magic-squa es
[5] H. Whi e, Bo de ed Magic Squa es - h p://budshaw.ca/Download.h ml
[6] W.S. And ews, Magic squa es and Cubes, Do e Publica ions, New Yo k
[7] Inde J. Taneja, Block-Wise Equal Sums Pandiagonal Magic Squa es o O de 4k, Zenodo, Janua y 31, 2019, pp. 1-17,
h p://doi.o g/10.5281/zenodo.2554288.
[8] Inde J. Taneja, Magic Rec angles in Cons uc ion o Block-Wise Pandiagonal Magic Squa es, Zenodo, Janua y 31, 2019, pp. 1-49,
h p://doi.o g/10.5281/zenodo.2554520.
[9] Inde J. Taneja, Block-Wise Equal Sums Magic Squa es o O de s 3kand 6k,Zenodo, Feb ua y 1, 2019, pp. 1-55,
h p://doi.o g/10.5281/zenodo.2554895.
[10] Inde J. Taneja, Block-Wise Unequal Sums Magic Squa es, Zenodo, Feb ua y 1, 2019, pp. 1-52,
h p://doi.o g/10.5281/zenodo.2555260.
[11] Inde J. Taneja, Block-Wise Magic and Bimagic Squa es o O de s 12 o 36, Zenodo, Feb ua y 1, 2019, pp. 1-53,
h p://doi.o g/10.5281/zenodo.2555343.
55
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
[12] Inde J. Taneja, Block-Wise Magic and Bimagic Squa es o O de s 39 o 45, Zenodo, Feb ua y 2, 2019, pp. 1-73,
h p://doi.o g/10.5281/zenodo.2555889.
[13] Inde J. Taneja, Nes ed Magic Squa es Wi h Pe ec Squa e Sums, Py hago ean T iples, and Bo de s Diffe ences, Zenodo, June
14, 2019, pp. 1-59, h p://doi.o g/10.5281/zenodo.3246586.
[14] Inde J. Taneja, Symme ic P ope ies o Nes ed Magic Squa es, Zenodo, June 29, 2019, pp. 1-55,
h p://doi.o g/10.5281/zenodo.3262170.
[15] Inde J. Taneja, Gene al Sum Symme ic and Posi i e En ies Nes ed Magic Squa es, Zenodo, July 04, 2019, pp. 1-55,
h p://doi.o g/10.5281/zenodo.3268877.
[16] Inde J. Taneja, Bo de ed Magic Squa es Wi h O de Squa e Magic Sums, Zenodo, Janua y 20, 2020, pp. 1-26,
h p://doi.o g/10.5281/zenodo.3613690.
[17] Inde J. Taneja, F ac ional and Decimal Type Bo de ed Magic Squa es Wi h Magic Sum 2020. Zenodo, Janua y 20, 2020, pp.1-25.
h p://doi.o g/10.5281/zenodo.3613698.
[18] Inde J. Taneja, F ac ional and Decimal Type Bo de ed Magic Squa es Wi h Magic Sum 2021, Zenodo, Decembe 16, 2020, pp.
1-33, h p://doi.o g/10.5281/zenodo.4327333.
[19] Inde J. Taneja, Block-Wise and Block-Bo de ed Magic Squa es Wi h Magic Sum 2022, Zenodo, Decembe 28, 2021, pp. 1-38,
h ps://doi.o g/10.5281/zenodo.5807789
[20] Inde J. Taneja, Block-Bo de ed Magic Squa es o P ime and Double P ime Numbe s - I, Zenodo, Augus 18, 2020, pp. 1-81,
h p://doi.o g/10.5281/zenodo.3990291.
[21] Inde J. Taneja, Block-Bo de ed Magic Squa es o P ime and Double P ime Numbe s - II, Zenodo, Augus 18, 2020, pp. 1-90,
h p://doi.o g/10.5281/zenodo.3990293.
[22] Inde J. Taneja, Block-Bo de ed Magic Squa es o P ime and Double P ime Numbe s - III, Zenodo, Sep embe 01, 2020, pp. 1-93,
h p://doi.o g/10.5281/zenodo.4011213.
[23] Inde J. Taneja, Block-Wise and Block-Bo de ed Magic and Bimagic Squa es o O de s 10 o 47. Zenodo, Janua y 14, 2021, pp.
1-185, h p://doi.o g/10.5281/zenodo.4437783.
[24] Inde J. Taneja, New Concep s in Magic Squa es: Double digi s Bo de ed Magic Squa es o O de s 7 o 108, Zenodo, Augus 09,
2023, pp. 1-30, h ps://doi.o g/10.5281/zenodo.8230214.
[25] Inde J. Taneja, Co ne ed Magic Squa es o O de 6, Zenodo, May 23, 2023, pp. 1-23, h ps://doi.o g/10.5281/zenodo.7960679.
[26] Inde J. Taneja, Co ne ed Magic Squa es o O de s 5 o 13, Zenodo, June 03, 2023, pp. 1-71,
h ps://doi.o g/10.5281/zenodo.8000467.
56
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-58, h ps://doi.o g/10.5281/zenodo.17330815
[27] Inde J. Taneja, New Concep s in Magic Squa es: Co ne ed Magic Squa es o O de s 5 o 81, Zenodo, Augus 09, 2023, pp. 1-27,
h ps://doi.o g/10.5281/zenodo.8231157.
[28] Inde J. Taneja, Re lexi e Yea 25: Ma hema ics o 25 and 2025 in Numbe s and Magic Squa es, Zenodo, Decembe 20, 2024, pp.
1-94, h ps://doi.o g/10.5281/zenodo.14533193.
[29] Inde J. Taneja, Numbe s and Magic Squa es Rep esen a ions o Ha dy-Ramanujan Numbe -1729, Zenodo, Decembe 20, 2024,
pp. 1-127, h ps://doi.o g/10.5281/zenodo.14538297.
[30] Inde J. Taneja, Magic Squa es o O de s 3 o 7 Rep esen ing Da es and Days o he Yea 2025, Zenodo, May 04, 2025, pp. 1-474,
h ps://doi.o g/10.5281/zenodo.15338142.
[31] Inde J. Taneja, Magic Squa es o O de 8 Rep esen ing Days and Da es o he Yea 2025, Zenodo, May 04, 2025, pp. 1-134,
h ps://doi.o g/10.5281/zenodo.15338246.
[32] Inde J. Taneja, Magic Squa es o O de 9 Rep esen ing Days and Da es o he Yea 2025, Zenodo, May 09, 2025, pp. 1-132,
h ps://doi.o g/10.5281/zenodo.15375349.
[33] Inde J. Taneja, Magic Squa es o O de 10 Rep esen ing Days and Da es o he Yea 2025, Zenodo, May 21, 2025, pp. 1-59,
h ps://doi.o g/10.5281/zenodo.15481738.
[34] Inde J. Taneja, Magic Squa es o o de 11 Rep esen ing Days and Da es o he Yea 2025, Zenodo, June 02, 2025, pp. 1-111,
h ps://doi.o g/10.5281/zenodo.15576562.
[35] Inde J. Taneja, Magic Squa es o o de 12 Rep esen ing Days and Da es o he Yea 2025, Zenodo, June 10, 2025, pp. 1-43,
h ps://doi.o g/10.5281/zenodo.15631884.
[36] Inde J. Taneja, Reduced En ies Magic and Semi-Magic Squa es o O de s 3, 5, 7 and 9, Zenodo, July 01, 2025, pp. 1-65,
h ps://doi.o g/10.5281/zenodo.15783321.
[37] Inde J. Taneja, Reduced En ies Magic and Semi-Magic Squa es o O de s 4, 6, 8 and 10, Zenodo, July 05, 2025, pp. 1-85,
h ps://doi.o g/10.5281/zenodo.15814675.
[38] Inde J. Taneja,Sel -Made Algeb aic Magic, Semi-Magic and Pandiagonal Magic Squa es o O de s 3 o 7, Zenodo, Sep embe 29,
2025, pp. 1-59, h ps://doi.o g/10.5281/zenodo.17219769.
[39] Inde J. Taneja, Sel -Made Algeb aic Magic, Semi-Magic and Pandiagonal Magic Squa es o O de 8, Zenodo, Sep embe 23, 2025,
pp. 1-65, h ps://doi.o g/10.5281/zenodo.17186001.
[40] Inde J. Taneja, Sel -Made Algeb aic Magic, Semi-Magic and Pandiagonal Magic Squa es o O de 9, Zenodo, Augus 27, 2025,
pp. 1-92, h ps://doi.o g/10.5281/zenodo.16955571.
57
